$A$ point moves in the $xy$-plane according to the following equations: $x = a \sin \omega t$ and $y = a(1 - \cos \omega t)$, where $a$ and $\omega$ are positive constants. Find the angle between the point's velocity and acceleration vectors.

Which of the following graphs represents the relationship between the height $(h)$ of a projectile and time $(t)$,when it is projected from the ground?

The real force $F$ acting on a particle of mass $m$ performing circular motion acts along the radius of a circle of radius $r$ and is directed towards the center of the circle. If the square root of the magnitude of such force is given by $\sqrt{F} = \frac{2 \pi}{T} \sqrt{m r}$,where $T$ is the periodic time,find the expression for the force $F$.

Fill in the blanks:
$(a)$ For vectors $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = \hat{i} + \hat{j} - 2\hat{k}$,the $y$-component of the vector $(\vec{A} - \vec{B})$ is .....
$(b)$ The formula for the total time of flight of a projectile projected vertically upwards with an initial velocity $u$ is ..........
$(c)$ The formula for the centripetal acceleration of a particle moving in a circular path of radius $R$ with a constant angular velocity $\omega$ is ......
$(d)$ The component of any vector is always .........

The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as a function of $\phi$ can be represented as:

$A$ train is moving along a straight line with a constant acceleration '$a$'. $A$ boy standing in the train throws a ball forward with a speed of $10 \ m/s$,at an angle of $60^{\circ}$ to the horizontal. The boy has to move forward by $1.15 \ m$ inside the train to catch the ball back at the initial height. The acceleration of the train,in $m/s^2$,is